Proton-conducting oxide materials are interesting objects from both fundamental and applied viewpoints due to the origination of protonic defects in a crystal structure as a result of their interaction with hydrogen-containing atmospheres at elevated temperatures. The high mobility of such defects at temperatures between 400 and 700 °C leads to superior ionic conductivity. As a result, some perovskite-type proton-conducting oxides have been proposed as electrolytes for solid oxide fuel and electrolysis cells. Barium cerate (BaCeO3), barium zirconate (BaZrO3), and barium cerate-zirconates (BaCeO3–BaZrO3) have been widely studied in terms of the parent phases of proton-conducting electrolytes. Among them, Y and Yb co-doped Ba(Ce,Zr)O3 can be identified as one of the most promising systems so far. This review discloses key functional properties of such phases and explains the increased attention of researchers to this system.